Talk to the renters landlord about the incident.
Let them know that their renter's behavior, not keeping his dog under control which entered your property to harass your chickens and shooting activity created a calamity.
Inquire of the police what options are available. If you don't want to do it directly, look for one or two when they go for lunch at a diner or such. Ask for advice and offer to buy their meal or coffee. If they are in the middle of their meal wait until after they finish.
Consult a lawyer who deals with this type of situation, they may know some ways to remedy this that you might not have thought of.
The neighbor who lost the dog, offer to replace their dog with any one of their choosing.
Now for your dog eating your chickens because the other is keeping him from eating. This is a dominance issue. You and your family need to be the assertive dominant pack leaders. **Do not consult the knucklehead on the back of your property line that thinks he's Cesar Milan. Very few of us will ever achieve what Cesar has achieved, not because we can't but because we don't dedicate ourselves as Cesar does.
Cesar offers great advice on feeding dogs and that is a great time to assert your dominance.
1. Always prepare the food in front of them.
2. Have separate bowls for each dog.
3. Keep them at a distance while preparing, outside an invisible boundary you mandate.
4. They must remain calm or they don't get their meal.
5. The one who is the most calm eats first.
6. Bite your dogs on the neck or hind quarters when they get out of line. (With your Shaolin Tiger Claw, not teeth.) This is how pack leaders assert dominance.
7. When they finish they can't approach another dogs bowl. If they do they get a bite (Tiger claw). They must move away from the remaining feeding dogs.
Cesar's Mastering Leadership series, especially Volume 2: Becoming a Pack Leader, will help you with your situation.
Vol. 2: Becoming a Pack Leader. Cesar shows you how to become your dog's pack leader, how to use your energy and body language to control your dog, and how to set rules and boundaries.
I'm no Cesar Milan but I'm trying.
Cheers